One of 3 item types in the TACHS Ability subtest
Figure matrices: a grid with one cell missing, and two rules that both fill it
Below: the directions as Riverside Assessments prints them, what its answer key says the rules turn on, and 3 real questions from the 960 of this type here — shown without their answers, because the answer comes back on the practice page after you choose.
What the directions say
“Find the figure that completes the puzzle.”
Riverside Assessments — TACHS Student Handbook, Ability sample directions
What the publisher’s answer key says the rules are
The handbook prints a handful of samples and explains them. Those explanations are the only published statement of what this item type turns on, so every question here is built to that vocabulary and nothing else.
“The figures in each matrix follow rules. One rule describes how the figures in every row are related. Another rule describes how the figures in every column are related.”
“Using either the rule for the rows or the rule for the columns, the missing shape is…”
“…the rule for every row is to add a triangle to the previous figure, moving in a clockwise direction. There is a different rule for the columns.”
“Look at the first and second rows. How are the figures in each row related? … Look at the first two columns.”
How to work one
- Step 1
Read the top row left to right
What changes from the first figure to the second, and from the second to the third? One thing — a shape added or taken away, a fill, a size, a turn. Check the second row does the same thing. That is the row rule.
- Step 2
Apply it to the third row
Take the third row's second figure and make the same change. The figure you get is the answer, and it must be among the five choices; if it is not, the change you named is not the rule.
- Step 3
Confirm down the column, or start there
The third column has its own rule, read off the first two columns. It reaches the same figure. Use it to check — or, if the row rule was hard to see, use it instead.
3 real questions, easiest first
Each is an original question of ours, lettered the way the TACHS letters its answer sheet. The answers are not on this page: pick one, then find out on the practice page, where every wrong choice explains why it is wrong.
Find the figure that completes the puzzle.
- A
- B
- C
- D
- E
Rows first. Look along the top row and name the one thing that changes from cell to cell. Check the middle row changes in the same way. Then take the bottom row’s middle figure and make that change once more — that is the figure to look for among the five.
How much moves. This one has one attribute changing in the easier direction, and it is size — the plainest change there is to see. Either direction reaches the answer, and the practice page will say which it was after you choose.
Find the figure that completes the puzzle.
- A
- B
- C
- D
- E
Rows first. Look along the top row and name the one thing that changes from cell to cell. Check the middle row changes in the same way. Then take the bottom row’s middle figure and make that change once more — that is the figure to look for among the five.
How much moves. This one has one attribute changing in the easier direction — a shape, a fill, a count or a turn — so name it and apply it. Either direction reaches the answer, and the practice page will say which it was after you choose.
Find the figure that completes the puzzle.
- J
- K
- L
- M
- N
Rows first. Look along the top row and name the one thing that changes from cell to cell. Check the middle row changes in the same way. Then take the bottom row’s middle figure and make that change once more — that is the figure to look for among the five.
How much moves. This one has two attributes changing together in each direction, so name both before you apply them, and check the other axis agrees. Either direction reaches the answer, and the practice page will say which it was after you choose.
Figure matrices questions here
Ability questions in all
of them free, no sign-up
item types in the subtest
Try ten, and see the answer after each one
The practice page deals ten Ability questions at a time — all 3 item types mixed, as the paper mixes them — marks each one the moment you choose, and says what the rule was. No sign-up and no email.
The HSPT has no Ability subtest, so this page is about the TACHS only.
Related questions
What is a figure matrices question?
A grid of figures, three across and three down, with the bottom-right cell empty. One rule relates the figures along every row and a different rule relates them down every column, and the child chooses, from five answer choices, the figure that completes the puzzle. Riverside Assessments’s key says the missing figure can be reached using either rule on its own.
How do you solve a figure matrix?
Look at the two complete rows and ask what changes from the first figure to the second to the third — a shape added, a fill changed, a turn. Apply that same change to the third row and you have the answer. If the row rule is hard to see, the columns carry a different rule that reaches the same figure, and that is the publisher’s own advice: either one is enough.
Is a figure matrices test the same as a spatial reasoning test?
It is one kind. “Spatial reasoning” is what prep sites call the whole of the TACHS Ability subtest, which the handbook describes as reasoning “with somewhat more novel questions that use spatial and figural content”. That subtest is three item types — Figure Matrices, Paper Folding and Figure Classification — and this page is about the first.
Is figure matrices on the HSPT?
No. The HSPT has no Ability subtest and no figure-based items; its five subtests are Verbal Skills, Quantitative Skills, Reading, Mathematics and Language. Figure Matrices is a TACHS item type.
Every quotation on this page is from the TACHS Student Handbook, which Riverside Assessments publishes. It is named here, with what it settles.